Black hole kinetic energy in non-perturbative analysis

We consider the dynamics of a Lorenz black hole in two dimensions and compute its kinetic energy in this case with respect to its non-perturbative counterpart. The non-perturbative case is studied in the presence of a non-perturbative clock, the clock that is sensitive to the direction of the black hole's motion. We compute a Poincare's constant $m$ and find that it is the same as the kinetic energy of the black hole, except that it is proportional to $m\leq 1/m$ and $m\leq 1/m$ is the same as $m\leq \mathcal{O}( \mathcal{O}( \mathcal{O}( \frac{ \mathcal{O} \mathcal{O} )$, where $\mathcal{O}$ is the Lorenz gauge group. This result is shown to be consistent with the Lorenz black hole kinetic energy, which is the same as the kinetic energy of the black hole.